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Vorticity inversion and actionatadistance instability in stably stratified shear flow

A. RABINOVICH, O. M. UMURHAN, N. HARNIK, F. LOTT and E. HEIFETZ

Journal of Fluid Mechanics / Volume 670 / March 2011, pp 301 325DOI: 10.1017/S002211201000529X, Published online: 14 January 2011

Link to this article: http://journals.cambridge.org/abstract_S002211201000529X

How to cite this article:A. RABINOVICH, O. M. UMURHAN, N. HARNIK, F. LOTT and E. HEIFETZ (2011). Vorticity inversion and actionatadistance instability in stably stratified shear flow. Journal of Fluid Mechanics, 670, pp 301325 doi:10.1017/S002211201000529X

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J. Fluid Mech. (2011), vol. 670, pp. 301–325. c© Cambridge University Press 2011doi:10.1017/S002211201000529X

301

Vorticity inversion and action-at-a-distanceinstability in stably stratified shear flow

A. RABINOVICH1, O. M. UMURHAN2,3, N. HARNIK1,F. LOTT4 AND E. HEIFETZ1†

1Department of Geophysics and Planetary Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel2School of Mathematical Sciences, Queen Mary, University of London,

London E1 4NS, UK3Department of Astronomy, City College of San Francisco, San Francisco, CA 94112, USA

4Laboratoire de Météorologie Dynamique, École Normale Supérieure, 24 rue Lhomond,75231 Paris CEDEX 05, France

(Received 17 February 2010; revised 24 August 2010; accepted 5 October 2010;

first published online 14 January 2011)

The somewhat counter-intuitive effect of how stratification destabilizes shear flowsand the rationalization of the Miles–Howard stability criterion are re-examined inwhat we believe to be the simplest example of action-at-a-distance interaction between‘buoyancy–vorticity gravity wave kernels’. The set-up consists of an infinite uniformshear Couette flow in which the Rayleigh–Fjørtoft necessary conditions for shear flowinstability are not satisfied. When two stably stratified density jumps are added, theflow may however become unstable. At each density jump the perturbation can bedecomposed into two coherent gravity waves propagating horizontally in oppositedirections. We show, in detail, how the instability results from a phase-locking action-at-a-distance interaction between the four waves (two at each jump) but can aswell be reasonably approximated by the interaction between only the two counter-propagating waves (one at each jump). From this perspective the nature of theinstability mechanism is similar to that of the barotropic and baroclinic ones. Nextwe add a small ambient stratification to examine how the critical-level dynamics altersour conclusions. We find that a strong vorticity anomaly is generated at the criticallevel because of the persistent vertical velocity induction by the interfacial wavesat the jumps. This critical-level anomaly acts in turn at a distance to dampen theinterfacial waves. When the ambient stratification is increased so that the Richardsonnumber exceeds the value of a quarter, this destructive interaction overwhelms theconstructive interaction between the interfacial waves, and consequently the flowbecomes stable. This effect is manifested when considering the different action-at-a-distance contributions to the energy flux divergence at the critical level. The interfacial-wave interaction is found to contribute towards divergence, that is, towards instability,whereas the critical-level–interfacial-wave interaction contributes towards an energyflux convergence, that is, towards stability.

Key words: atmospheric flows, stratified flows

1. IntroductionRecently Harnik et al. (2008) developed a ‘buoyancy–vorticity wave interaction

approach’ to describe the linear dynamics of stably stratified shear flows. The essence

† Email address for correspondence: [email protected]

302 A. Rabinovich, O. M. Umurhan, N. Harnik, F. Lott and E. Heifetz

Initial vorticityanomaly

–q

+q

x

z

ρ0 – |�ρ–|

ρ0 + |�ρ–|

ρ0

Action-at-a-distance

Vorticity generation

Action-at-a-distance

(a) (b)

(c) (d)

Figure 1. The evolution of perturbations on a fluid of three density layers at time (a, c) t0and (b, d ) t0 + �t . An initial vorticity anomaly on an interface between two density layers(c) induces a far field velocity which perturbs other interface levels (a). These new displacement(buoyancy) anomalies generate vorticity locally (b) with an action-at-a-distance effect as well(d ). As a result, a new material perturbation is formed, altering the initial vorticity anomaly.The large arrows represent the local vertical velocity resulting from the vorticity anomaly,while the small dashed arrows are the vertical velocity induced by a remote layer.

of their view is that in stably stratified flows the buoyancy acts as a restoring force,and therefore a horizontal (say zonal) gradient of material displacement generatesvorticity in the meridional direction. By inversion such an initial localized vorticityanomaly at some particular layer induces a non-local vertical velocity field whichdeforms the flow at remote layers, generating vorticity in the far field. The latter,in turn, induces a far-field velocity which deforms the initially perturbed layer andconsequently alters the initial vorticity anomaly (figure 1). Thus, even in the absenceof a mean vorticity gradient, vorticity inversion and action-at-a-distance are at theheart of stratified shear flow dynamics.

In barotropic and baroclinic shear flows the basic (potential) vorticity buildingblocks which interact at a distance are the counter-propagating Rossby wavesor CRWs (Bretherton 1966; Hoskins, Mcintyre & Robertson 1985). In stratifiedshear flows the basic building blocks at each level are mixed gravity–Rossby-wavekernels that, in the absence of shear curvature, are reduced to two oppositelypropagating internal gravity wave vortex sheets. The fact that at each given leveltwo building blocks exist (rather than one CRW kernel in the barotropic/barocliniccase) complicates the dynamics; however this is a direct consequence of the interplaybetween the buoyancy and vorticity fields.

The two necessary conditions for modal instability of barotropic and baroclinicshear flows are straightforwardly rationalized from the CRW perspective. Mutualamplification between two CRWs are possible only if the mean vorticity gradient

Vorticity inversion and action-at-a-distance instability 303

Z = h

Z = –h

U–(z) U

–(z) U

–(z)

(a) (b) (c)

ρ–(z) ρ–(z) ρ–(z)

ρ–, U–

z

Figure 2. The mean density and shear profiles (thick black contours and grey contours witharrows respectively) of the models analysed in this work: (a) two density jumps; (b) a similar,yet continuous, density profile (see Appendix A); and (c) two density jumps with a smallambient density gradient.

at their vicinities changes sign – the barotropic Rayleigh (1880) and the baroclinicCharney & Stern (1962) conditions. The Fjørtoft (1950) condition states that themean vorticity gradient should be positively correlated with the mean velocity profile,as measured with respect to the point of zero vorticity gradient. When this conditionis satisfied the CRWs are able to counter-propagate against the shear, which allowsthem to phase lock in their constructive interaction configuration. It is thus easyto understand why these conditions may be irrelevant to a stratified shear flow.The mechanism of vorticity anomaly generation by horizontal buoyancy gradientsis independent of the presence of mean vorticity gradients. Furthermore, at everylayer one of the two gravity wave kernels (propagating in opposite directions inthe mean flow frame of reference) is always counter-propagating against the meanflow. Moreover, the Rayleigh and Fjørtoft conditions result, respectively, from theglobal conservation of pseudo-momentum and pseudo-energy (Shepherd 1990). Theequivalent Miles–Howard (M–H; Howard 1961; Miles 1961) necessary condition forstratified shear flow instability (that the Richardson number should be smaller thana quarter somewhere within the domain) does not seem to result from an eddy-associated global conservation law, and therefore it is more challenging to interpret.One of the main aims of this study is to gain a deeper insight on this condition fromthe kernel gravity wave (KGW) interaction standpoint.

The mechanistic picture of the simple barotropic Rayleigh (1880) and the baroclinicEady (1949) models, in terms of a two-CRW interaction (Davies & Bishop 1994;Heifetz et al. 1999), served as a guiding line in much more complex shear flows(Heifetz et al. 2004), including the nonlinear eddy life cycles on realistic jets (Methvenet al. 2005). Here we follow the same route and examine in detail an equivalent simplemodel for a stratified shear flow of two density jumps in the presence of constantshear (figure 2a). This set-up, examined already by Taylor (1931) and Goldstein(1931), is even simpler than the model of Holmboe (1962), often invoked to address


basic questions in stratified shear flows. In some senses, the following analysis issimilar to those done in the previous works on shallow water by Hayashi & Young(1986), Sakai (1989) and Knessl & Keller (1992). In their models the instability is alsorationalized in terms of action-at-a-distance interaction between localized counter-propagating waves, when the Rayleigh criterion is not satisfied. After formulatingthe problem and briefly reviewing the buoyancy–vorticity KGW interaction approach(§ 2) we analyse, in detail, the four-KGW interaction of this problem (§ 3). As expectedthe instability is dominated by the two counter-propagating kernels; however for longwavelengths the other two pro-propagating waves, which must be hindered, contributeas well to the solution.

The common interpretation for the M–H condition stems from the over-reflectionperspective, which is based on a cross-shear wave propagation view. When analysingthe index of refraction of wave propagation across the shear in the Taylor–Goldsteinequation Lindzen & Barker (1985) found that the wave geometry which allows over-reflection, and hence instability, must include a critical level (hereafter CL) whichseparates between the wave propagation and wave evanescence regions. To obtainan evanescence region at the vicinity of a CL one must have local values of theRichardson number which are smaller than a quarter (Booker & Bretherton 1967).Alternatively, Baines & Mitsudera (1994; see also Baines 1998, chapter 4) used awave interaction approach, similar to our KGW interaction, but they interpretedthe suppression of the instability in terms of absorption of the two waves at theCL between them (a cross-shear wave propagation view). Here we interpret thisstabilizing effect of the CL purely in terms of counter-propagating kernel–waveinteraction.

The M–H condition is obviously satisfied in the simple set-up of the two densityjumps, since the Richardson number is concentrated in two delta functions there(a continuous density profile with two sharp density gradients reveals the samebehaviour, as shown in Appendix A). When increasing slightly the stratificationelsewhere so that the ambient Richardson number increases from zero to a quarter,the instability is suppressed, in agreement with the M–H condition. Nonetheless, itseems counter-intuitive that such weak ambient stratification (a normalized value ofone quarter compared with delta functions) will dominate over the dynamics of thedensity jumps. In § 4 we analyse the destructive effect of the CL consistently fromthe buoyancy–vorticity action-at-a-distance approach. It is shown how the KGWs atthe density jumps generate vorticity at the CL which acts to mask the constructiveinterference between the KGWs in the far field. We discuss these results in § 5.

2. KGW formulation for stratified shear flowWe consider an inviscid incompressible Boussinesq two-dimensional flow in the

zonal–vertical (x–z ) plane, with a zonally uniform basic state that varies only withheight and is in hydrostatic balance. For this set-up the linearized momentum andcontinuity equations become

Du

Dt= −Uzw −

1

ρ0

∂p

∂x, (2.1a)

Dw

Dt= b − 1

ρ0

∂p

∂z, (2.1b)


+q +q

+q +q

–q

–q –q

–q

4T

t = t0 +

q ~ ς t = t0

Figure 3. The gravity wave mechanism for a ζ+ wave in a single-jump configuration. Thewavy contour represents the displacement, while the positive and negative vorticities are shownwith the regular and dashed circles respectively. Notice that the vorticity and the displacementare in phase as the KGW formulation prescribes for the ‘plus’ kernel. The top wave is fora random time, t= t0, and the bottom wave is the same wave after a fourth of its period,t = t0 +T/4. The dashed thick arrows show the buoyancy force on the wave, which is simply thetendency of the disturbances to level out to equilibrium. The solid filled circles mark the axispoints which rotate in the direction of the circular arrows as a result of the levelling motion.These points have zero vorticity at t= t0 but generate the most vorticity with time and arealso the points of maximum vertical velocity marked with the thin long arrows. Taking intoconsideration all of the processes in the scheme – interface displacement due to the verticalvelocity and the flattening accompanied by generation of new vorticity – accounts for therightward phase propagation depicted.

Db

Dt= −wN2, (2.1c)

∂u

∂x+

∂w

∂z= 0, (2.1d )

where U is the mean zonal velocity with the mean shear Uz taken to be constantin this study. The linearized material derivative is D/Dt = (∂/∂t) + U (∂/∂x); (u, w)are the perturbation velocities in the x and z directions and p is the perturbationpressure; ρ0 is a constant reference density; b = −(ρ/ρ0)g is the perturbation buoyancy,where ρ is the density perturbation and g the gravity. The Brunt–Väisälä frequencyN2 = −(g/ρ0)(dρ/dz) = bz is the mean vertical buoyancy gradient, where ρ is the meandensity profile.

Equation set (2.1) yields, after some algebra, the vorticity equation

Dq

Dt=

∂b

∂x= −bz

∂ζ

∂x, (2.2)

where q = (∂w/∂x) − (∂u/∂z) is (minus) the perturbation component of the vorticityperpendicular to the zonal–vertical plane and ζ is the vertical displacement.Equation (2.2) simply states that since the buoyancy is a restoring force its tendencyto flatten the deformed isopycnals back generates vorticity to the right of theperturbation buoyancy gradient (figure 1; see also figure 3). By inversion, q induces afar-field vertical velocity that generates vertical displacement which, in turn, generatesvorticity in the far field. To formulate this mathematically we look at a zonal Fouriercomponent of the form of eikx , with wavenumber k, and introduce its streamfunction


ψ(k, z) satisfying u = −∂ψ/∂z, w = ikψ and q = −k2ψ + ∂2ψ/∂z2. Then the verticalvelocity at some height z can be expressed in terms of the integrated contribution ofthe far field vorticity field,

w(z) =

∫z′

q(z′)G(z, z′) dz′, (2.3)

where G(z, z′) is the Green function satisfying −k2G + (∂2G/∂z2) = ikδ(z − z′) withthe appropriate boundary conditions. Here we apply the formulation to open flowwhose perturbation vanishes at ±∞ so that G(z, z′) = −(i/2) exp(−k|z − z′|). Writingthen

Dζ

Dt= w =

∫z′

q(z′)G(z, z′) dz′ (2.4)

formulates, together with (2.2), the mathematical description for the action in adistance interaction between the buoyancy b = −bzζ and the vorticity q .

The simplest non-trivial set-up of vorticity–buoyancy interaction is of a single stablystratified density jump:

ρ = ρ0 − |�ρ|H (z), (2.5)where H (z) is the Heaviside function and the density interface is at z = 0. Thecorresponding mean buoyancy gradient is then

bz = −g

ρ0

dρ

dz=

g|�ρ|ρ0

δ(z) = �bδ(z). (2.6)

For finite vertical displacement ζ , (2.2) implies that the vorticity perturbationshould be a delta function as well, q = q̂(x, t)δ(z). Equations (2.2) and (2.4) can thenbe written in the matrix form:

∂

∂t

(q̂

ζ

)= −ik

⎛⎝U 0 �b12k

U 0

⎞⎠( q̂ζ

), (2.7)

where U 0 is the mean velocity at z = 0. The solution of this simple eigenproblem isa superposition of the two Boussinesq internal gravity waves with the intrinsic phasespeeds:

c± = ±

√�b

2k≡ ±cgr , (2.8)(

q̂

ζ

)= ζ+0

(2kcgr

1

)exp[−ik(cgr + U 0)t] + ζ −0

(−2kcgr1

)exp[−ik(cgr + U 0)t], (2.9)

where ζ ±0 are the vertical displacements at z =0 of the two modes. Hence, the vorticityand the vertical displacement of the eastward-propagating (westward-propagating)wave, in the mean flow frame of reference, are in phase (anti-phase), ensuring coherentpropagation of the buoyancy and the vorticity fields (figure 3). Next, we use thesetwo waves as KGW building blocks, to examine the interaction between two densityjumps (see Harnik et al. 2008 for more details on the KGW formulation).

3. Two-density-jump instability3.1. Formulation

We consider the density profile of two density jumps located at z = ±h (figure 2a):

ρ = ρ0 + |�ρ|[1 − H (z − h) − H (z + h)]. (3.1)


Normalizing the length by h and the time by 1/Uz, (2.2) and (2.4) become(∂

∂t+ ikz

)q = −ikRiζ, (3.2a)

(∂

∂t+ ikz

)ζ = − i

2

∫z′

q(z′) exp(−k|z − z′|) dz′, (3.2b)

where all variables are non-dimensionalized and Ri = bz/(Uz)2 is the Richardson

number. For the two-density-jump profile

Ri =�b

h(Uz)2[δ(z − 1) + δ(z + 1)] ≡ R̂i[δ(z − 1) + δ(z + 1)]. (3.3)

The M–H condition, although commonly related to a continuously stratifiedflow, is obviously satisfied in this simple set-up. Taking the normalized M–Hintegral (cf. (4.3.3) in Baines 1998) the necessary condition to obtain non-zero

imaginary phase speed requires that∫

(|φz|2 + k2|φ|2) dz + R̂i[(|φ|2/|z − c|2)z=−1 +(|φ|2/|z − c|2)z=1] = (1/4)

∫(|φ|2/|z − c|2) dz, where φ =ψ/

√(z − c). Since all the terms

on the two sides of the equation are positive definite the condition can be satisfied.Nonetheless, for completeness, we analyse (in Appendix A) a continuous smoothdensity profile (figure 2b) whose limit converges to (3.1). Denoting hereafter levelz = 1 with the subscript 1 and z = −1 with 2 so that

q = q̂1δ(z − 1) + q̂2δ(z + 1), (3.4)

we write (3.2a,b) explicitly for the two jumps:(∂

∂t+ ik

)q̂1 = −ikR̂iζ1, (3.5a)(

∂

∂t+ ik

)ζ1 = −

i

2(q̂1 + q̂2 e

−2k), (3.5b)(∂

∂t− ik

)q̂2 = −ikR̂iζ2, (3.5c)(

∂

∂t− ik

)ζ2 = −

i

2(q̂1 e

−2k + q̂2). (3.5d )

And the intrinsic gravity wave speed in (2.8) is normalized to cgr =

√R̂i/2k. By

definition the two KGWs at each interface must preserve their eigenstructure of (2.9):

q̂±1/2 = ±

√2kR̂iζ ±1/2, (3.6)

where, as before, the superscript ± indicates the sign of the KGW-associated intrinsicpropagation speed. Since the vorticity and the displacement at each interface aresuperpositions of the two KGWs (q̂1/2 = q̂

+1/2 + q̂

−1/2, ζ1/2 = ζ

+1/2 + ζ

−1/2), (3.5a–d ) can be

rearranged to be written as an equation set for the KGW displacements:[∂

∂t+ ik(cgr + 1)

]ζ+1 = −iσ

(ζ+2 − ζ −2

), (3.7a)

[∂

∂t+ ik(−cgr + 1)

]ζ −1 = −iσ

(ζ+2 − ζ −2

), (3.7b)

308 A. Rabinovich, O. M. Umurhan, N. Harnik, F. Lott and E. Heifetz[∂

∂t+ ik(cgr − 1)

]ζ+2 = −iσ

(ζ+1 − ζ −1

), (3.7c)

[∂

∂t+ ik(−cgr − 1)

]ζ −2 = −iσ

(ζ+1 − ζ −1

). (3.7d )

Equation set (3.7a–d ) is straightforward to interpret. The mean normalized velocity atthe interfaces is U (±1) = ±1; hence in the absence of interaction with the KGWs of theopposed interface (when the right-hand side are zeros), ζ+1 and ζ

−2 tend to propagate in

the direction of the mean flow whereas ζ −1 and ζ+2 propagate counter to it. Each KGW,

by construction, does not interact with the other KGW of the same interface, but with

the two KGWs of the opposed one. The interaction coefficient σ ≡ (e−2k/4)√

2kR̂i,results from the vorticity–displacement production relation (3.6) and the exponentialevanescence of the induced vertical velocity as represented by the Green function,where the additional division by the factor 2 results from the equipartition of thevorticity at each interface between the two KGWs. The −i prefix on the right-handside indicates that the vertical velocity is induced a quarter wavelength to the rightof the vorticity, recalling that for ζ+ the vorticity and the displacement are in phase,whereas for ζ − they are anti-phased (hence the minus sign before ζ −; (3.6)).

3.2. Analysis and results

The eigenvalue solution of (3.7), for normal modes of the form of exp[ik(x − ct)],yields the dispersion relation

c = ±

√c2gr + 1 ± 2cgr

√1 +(σ

k

)2. (3.8)

Without interaction (σ =0) the solution is reduced to the four uncoupled modes:

c = ± (cgr ± 1). For σ �= 0 instability is obtained when 2cgr√

1 + (σ/k)2 >c2gr + 1.The unstable modes are stationary, cr = 0, experiencing a growth rate of kci = k ·√

−(c2gr + 1) + 2cgr√

1 + (σ/k)2. Figure 4(a) shows the growth rate as a function of

(k, R̂i). As can be seen a global maximum for the growth rate is obtained when

(k, R̂i) ≈ (0.5, 1). The wavenumber range of instability shrinks and shifts to higherwavenumbers as R̂i increases. For R̂i smaller than unity only a short-wave cutoff

exists, whereas for R̂i larger than unity both short- and long-wave cutoffs exist. Theunstable and stable modes come in conjugate pairs. The growth rate of these modesas well as the real phase speed of the four modes are shown for representative values

of R̂i in figures 5(a) and 5(b).The amplitude ratios and the phase differences between the four KGWs for the

unstable modes (and their continuation to the neutral regions at the two sides ofthe cutoffs) are shown in figures 5(c) and 5(d ) (normalized with respect to ζ −1 ). In theunstable region the two counter-propagating waves (ζ −1 , ζ

+2 ) have equal amplitudes.

The pro-propagating waves (ζ+1 , ζ−2 ) are also of equal amplitude, smaller by a factor

χ (figure 4b). Each counter-propagating wave is aligned in anti-phase (in bothvorticity and displacement) with the pro-propagating wave of the opposed levelso that (ζ+1 , ζ

−2 ) = −χ(ζ+2 , ζ −1 ). The two counter-propagating waves are in phase in

their displacement (and anti-phased in their vorticity) at the long-wave cutoffand


Amplitude ratio (χ)

Wavenumber (k)

Growth rate (kci)(a)

0.2 0.4 0.6 0.8 1.0

Wavenumber (k)

0.2 0.4 0.6 0.8 1.0

Wavenumber (k)

0.2 0.4 0.6 0.8 1.0

0.5

1.0

1.5

2.0

2.5 (b)

0.5

1.0

1.5

2.0

2.5 (c)

0.5

1.0

1.5

2.0

2.5(�2

+ – �1– )/π

0

0.02

0.04

0.06

0.08

0.10.20.30.40.50.60.70.8

0

0.2

0.4

0.6

0.8

1.0

Hindering

Helping

kci = 0.0956(0.46,1.05)

Ri

Figure 4. (a) Growth rate kci , (b) amplitude ratio (χ), Z+1 /Z

+2 =Z

−2 /Z

−1 , and (c) phase

difference between counter-/pro-propagating waves, �, as functions of R̂i and k. The twoblack curves mark the region of instability (the boundaries of the non-zero growth rate in a).The global maximum in growth rate is marked in (a).

−0.1

0

0.1

kci

kci

kci

kci

Ri = 2

Exact solution

−0.1

0

0.1Ri = 1.5

−0.1

0

0.1Ri = 1

0 0.5 1.0−0.1

0

0.1

Wavenumber (k)0 0.5 1.0

Wavenumber (k)0.5 1.0

Wavenumber (k)0 0.5 1.0

0.5 1.0 0 0.5 1.0

0.5 1.0 0 0.5 1.0

0.5 1.0 0 0.5 1.0

Wavenumber (k)

Ri = 0.5

−1

0

1

c rc r

c rc r

−1

0

1

−1

0

1

−1

0 0.5 1.0 0 0.5 1.0

0 0.5 1.0 0 0.5 1.0

0 0.5 1.0 0 0.5 1.0

0

1

0

123

−4−2

024

0

1

2

−4−2

024

ζ1

+ζ

2

+

0

1

2

−4−2

024

0

1

2

Am

plit

ude

Am

plit

ude

Am

plit

ude

Am

plit

ude

−4−2

024

Pha

se (�)

P

hase

(�)

P

hase

(�)

P

hase

(�)

(a) (b) (c) (d)χ = 0 approximation ζ2

–

Figure 5. Dispersion relation and wave structure as functions of k for different values ofR̂i. (a) The growth rate (kci) for the exact solution with four KGWs (solid lines) and theapproximate solution (χ = 0) with two KGWs (dashed lines). (b) The real part of the phasespeed for the exact solution. (c) The amplitude of the three kernels (for the exact solution)ζ

±2 and ζ

+1 measured against the wave ζ

−1 which is set with Z

−1 = 1 and

−1 = 0. (d ) The

corresponding phase of the three kernels. The values of R̂i are noted on the left side. Thevertical dashed lines in (c) and (d ) mark the instability region. As R̂i decreases, the rangeof instability shifts to smaller wavenumbers, where the (χ = 0) approximation is no longeradequate.


Level 1(z = 1)

Level 2(z = –1)

U–

U–

Counter-propagatingPro-propagating

Counter-propagating Pro-propagating

x

z

Figure 6. The four-KGW configuration of the most unstable normal mode of thetwo-density-jump problem (for which � = π/2). The displacement of the counter- andpro-propagating waves are denoted by the solid and dashed wavy lines respectively, and thecorresponding kernel vertical velocity is denoted by the solid and dashed arrows respectively.Notice that both the counter- and pro-propagating pairs are mutually amplifying, and eachpair hinders the other pair.

anti-phased in their displacement (and in phase in their vorticity) at the short-wavecutoff. In between, the upper counter-propagating wave displacement (vorticity) isshifted eastward (westward) with respect to the lower counter-propagating one. Aschematic of the KGW set-up for the case in which the counter-propagating wavesare in quadrature is drawn in figure 6. Next we wish to interpret these results fromthe KGW interaction point of view.

The first question that might arise is why the pro-propagating kernels are present atall in the growth mechanism. For comparison, in the analogous barotropic Rayleigh(1880) and baroclinic Eady (1949) models, only CRWs exist, and they are phaselocked to yield unstable modes, similar to the counter-propagating waves (ζ −1 , ζ

+2 )

of our set-up. The reason lies in the fundamentally different action-at-a-distancemechanism between these problems. In the barotropic/baroclinic cases the inducedvelocity field generates vorticity at a distance by advecting the local mean (potential)vorticity gradient so that the displacement and the vorticity anomalies are either inphase or anti-phased according to the sign of the mean vorticity gradient. In ourproblem, the induced velocity deforms the buoyancy in the far field, while vorticity isonly subsequently induced in quadrature, by the new horizontal buoyancy gradients.This initial pure buoyancy anomaly necessarily excites both KGWs. Furthermore, inorder to get growth the buoyancy (or displacement) field cannot be fully in phasewith or anti-phase to the vorticity field; hence they cannot be described solely by thecounter-propagating kernel. To see this mathematically we write the displacement andvorticity in terms of their amplitudes and phases (ζ =Z eiθ , q = Q eiα) and substitutea modal solution in the vorticity generation equation (2.2). Then by taking the ratio


between the imaginary and real parts of (2.2) we obtain

tan (θ − α) = cicr − U

. (3.9)

In our set-up tan (θ − α) = ∓ci for levels z = ±1 so that (everywhere outside the CL)ci �= 0 is obtained only when (θ − α) �= (0, π). Since the pro-propagating waves mustexist to obtain growth, but must also be phase locked, they must be hindered by thecounter-propagating waves of the opposed level. The induced velocity by the latter onthe former should be strong enough (hence the amplitude ratio χ) and of the oppositesign (hence the anti-phased alignment) to overcome the pro-propagation tendency ofthe former (figure 6). Since for the unstable modes (ζ+1 , ζ

−2 ) = −χ(ζ+2 , ζ −1 ), the four

KGW equations (3.7) can be reduced (without approximation) into two equations forthe two counter-propagating KGWs:[

∂

∂t+ ik(−c̃gr + 1)

]ζ −1 = −iσ ζ+2 , (3.10a)

[∂

∂t+ ik(c̃gr − 1)

]ζ+2 = iσ ζ

−1 , (3.10b)

where c̃gr = cgr − χ(σ/k) is the counter-propagating KGW intrinsic phase speed,reduced by the hindering effect of the anti-phased pro-propagating wave at theopposed boundary. Writing ζ −1 = Z

−1 e

i−1 , ζ+2 =Z+2 e

i+2 and substituting it in (3.10a,b),the real parts yield the instantaneous KGW growth rates,

Ż−1Z−1

=Z+2Z−1

σ sin (�),Ż+2Z+2

=Z−1Z+2

σ sin (�), (3.11a,b)

while the imaginary parts yield the instantaneous KGW phase speeds,

c−1 = −

̇−1k

= (1 − c̃gr ) +Z+2Z−1

σ

kcos�, c+2 = −

̇+2k

= −(1 − c̃gr ) −Z−1Z+2

σ

kcos �,

(3.12a,b)

where � = +2 −−1 . For modal growth rate, kci = Ż−1 /Z−1 = Ż+2 /Z+2 ; thus it is obviousthat the KGWs have equal amplitudes so that kci = σ sin (�). In order for the wavesto be phase locked, cr = c

−1 = c

+2 , the waves should be stationary with phase locking

�, satisfying cos (�) = (c̃gr − 1)/(σ/k). Thus, the KGW dynamics is just like the twoCRWs in barotropic and baroclinic instabilities, except that the intrinsic phase speedis proportional to the square root of the wavelength (rather than the wavelength itselffor the Rossby kernels) and is somewhat hindered by an opposed anti-phased kernel.

For small wavenumbers the single counter-propagation phase speed is large (2.8),and the KGWs need to be anti-phased in order to hinder each other’s counter-propagation (figure 4c). Furthermore, the opposed pro-propagating kernel amplitudeis also large (figure 4b) to amplify the hindering. Since the counter-propagation speed

is also proportional to R̂i, as it increases, such hindering can maintain phase locking

only for larger wavenumbers; hence a long-wave cutoff is established for R̂i > 1(figure 4a). As the wavenumber slightly increases, the additional hindering effectby the opposed anti-phased kernel allows the KGWs to be in a hindering–growingconfiguration (0 < � < π/2). Ignoring this effect (by taking χ = 0) prevents modalgrowth in this range (figure 5a). For larger wavenumbers the amount of hinderingrequired for phase locking is reduced, and the KGWs can be shifted towards a


more amplifying configuration, which becomes optimal when � = π/2 (figure 6).This occurs for wavenumbers and Richardson numbers that satisfy c̃gr = 1, whichis when each KGW (with its anti-phased opposed one) counter-propagates with aspeed which balances exactly the mean flow speed. For even larger wavenumbers thecounter-propagation speed is overwhelmed by the mean flow, and the KGWs needto be in a helping–growing configuration (π/2 < � < π). In this range the hinderingeffect of the pro-propagating waves interferes with the phase locking, and therefore χdecreases (figure 4b). Eventually the wavenumber becomes too large for effective phaselocking (because both the counter-propagation speed and the interaction coefficientσ decrease with the wavenumber), and a short-wave cutoff is established. Increasingthe Richardson number increases the counter-propagation speed and hence shiftsboth the long- and short-wave cutoffs to higher wavenumbers but also shrinks thetransition between them.

Summarizing this part of the analysis, we examine in detail the action-at-a-distanceparadigm of Baines & Mitsudera (1994) to stratified shear flow instability, in thesimplest model we could construct. This model shares many similarities with thebaroclinic Eady (1949) and barotropic Rayleigh (1880) models, but is fundamentallydifferent in the sense that it does not satisfy the Rayleigh–Fjørtoft conditions forinstability and that each interface supports two edge waves rather than one. Althoughthe dynamics is more complex, the essence of the instability is the same – it isdominated by the phase-locked constructive interaction between the two counter-propagating interfacial waves (IWs).

These three models, however, do not contain a CL (they do contain a steering levelin the middle, where U = cr , but the vorticity perturbation is zero there). Therefore,in the next section we proceed by adding a small ambient stratification to the twodensity jumps (figure 3c), to examine the effect of the CL on the instability from thevorticity–buoyancy action-at-a-distance perspective.

4. Suppression of instability by ambient stratification4.1. Results

We consider the density profile of figure 3(c):

ρ = ρ0 + |�ρ|[1 − H (z − h) − H (z + h)] − z|ρz|amb (4.1)

(|ρz|amb = constant), where the ambient Richardson number (outside the jumps)satisfies 0 < Riamb = −[(g/ρ0)|ρz|amb]/(Uz)2 < 1/4 so that

Ri(z) = Riamb + R̂i[δ(z − 1) + δ(z + 1)]. (4.2)

We find the normal-mode solutions of this set-up both by numerical discretizationof (2.2) and (2.4) (Appendix B) and by converting them to the Taylor–Goldsteinequation and solving it analytically (Appendix C). The agreement between the two

methods of solution is excellent (the two solutions for R̂i = 2, Riamb = 0.2 and k = 1.07differ at most by 0.46 %). As in the previous set-up four discrete normal modes exist,for each zonal wavenumber, where the growing and decaying modes (if they exist)come in conjugate pairs.

Figure 7 shows the dispersion relation for the growing and decaying modes asRiamb increases gradually from 0 to 1/4. We choose to show the case in which

R̂i = 2, since in the absence of ambient stratification the dispersion relation includesa distinct long-wave cutoff, and the role of the pro-propagating IWs is negligible (cf.


0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Wavenumber (k)

Gro

wth

rat

e (k

c i)

Riamb = 0.249Riamb = 0.24Riamb = 0.225Riamb = 0.2Riamb = 0.1Riamb = 0.05Riamb = 0

Figure 7. The growth rate (kci) as a function of wavenumber (k ) for different values ofRiamb given in the figure legend, with the outermost curve corresponding to Riamb = 0 and theinnermost curve to Riamb = 0.249. For all wavenumbers in the unstable zone there are twocomplex-conjugate phase speeds c = ± ici . The real part of the phase speed for the unstablemodes is zero in all cases.

the topmost row of figure 5; the pro-propagating IWs were found to be unessentialto the dynamics with ambient stratification). As Riamb increases the growth ratedecreases, and the spectrum emitting the instability shrinks and is shifted towardshigher wavenumbers. As expected from the M–H criterion the instability ceases whenthe Richardson number becomes larger than a quarter everywhere in the flow. Infigure 8 we show the structure of the amplitudes and phases of the vorticity and thevertical displacements, for a specific wavenumber (k = 1.07) and for various values ofRiamb. The IWs hardly change with Riamb; the vorticity IWs remain tilted against theshear, by approximately a quarter of wavelength, and the displacement of the lower(upper) IW is almost in phase (anti-phase) with its vorticity. Although the ambientstratification is constant everywhere in the domain the vorticity and the displacementare concentrated at the critical layer whose width decreases as the value of Riambapproaches a quarter (e.g. figure 8).

4.2. Interfacial-wave–critical-layer far-field interaction

This structure, which is drawn for Riamb = 0.249 in figure 9, can be understood fromthe vorticity–buoyancy view when considering first the vertical displacement profilein the absence of ambient stratification. Although the vertical velocity resultingfrom the combined induction of the two IWs is minimized at the CL (becauseof the evanescence of the Green function), the vertical displacement is maximizedthere. This is because the fluid at the CL is phase locked with the two IWs,and therefore its vertical displacement is amplified persistently ((2.4) yields thatζ = w/[ik(U − cr ) + kci] = w/kci at the CL) and tends to infinity as ci goes to zero.In the presence of ambient stratification the vertical displacement at the CL deformsthe isopycnal surfaces and hence generates strong vorticity a quarter wavelength


−1.5 −1.0 −0.5 0 0.5 1.0 1.50

0.2

0.4

0.6

0.8

1.0A

mpl

itud

e

−2 −1 0 1 2−0.5

0

0.5

1.0

1.5

2.0

Height

−1.5 −1.0 −0.5 0 0.5 1.0 1.5

−2 −1 0 1 2

Height

Pha

se/π

Riamb = 0

Riamb = 0.10

Riamb = 0.20

Riamb = 0.24

0

1

2

3

4

5(×10−3)

Riamb = 0

Riamb = 0.10

Riamb = 0.20

Riamb = 0.24

0.8

1.0

1.2

1.4

1.6

1.8

Riamb = 0Riamb = 0.10Riamb = 0.20Riamb = 0.24

−0.05 0 0.050

0.2

0.4(a)

(c)

(b)

(d) Riamb = 0Riamb = 0.10

Riamb = 0.20

Riamb = 0.24

Figure 8. (a,b) Amplitude and (c, d ) phase of (a, c) the vorticity and (b, d ) the displacement

as functions of height for parameter values R̂i = 2 and k = 1.07 and varying Riamb shown inthe figure legend. Values are normalized so that the maximum amplitude is 1 and the vorticityphase is 0 at z = 1. Note that all q amplitude curves (a) lie on top of each other at the jumps,and for Riamb = 0 the vorticity is zero except at the jumps; thus its phase is marked as dots in(c). In the critical layer, when Riamb > 0, the amplitudes are increased, and a rapid transitionof the phase occurs. Away from the critical layer, q and ζ are approximately in two regionsof equal phase.

to its right in a standing-wave-like fashion (q = i(Riamb/ci)ζ ; (2.2) and figure 9; cf.Appendix B in Harnik et al. 2008).

Adding a small background stratification thus adds a third active region to the two-jump dynamics. It would simplify things, conceptually, to represent the CL regionas a single standing-wave kernel with a structure at the exact centre of the CL(q and ζ in quadrature in both space and time; cf. Appendix B of Harnik et al.2008). Such a vorticity kernel induces an evanescent untilted vertical velocity field.This representation is valid, since the far-field vertical velocity induced by the entireCL (the region in which q and ζ are strongly phase tilted with height) is indeedevanescent and almost untilted (a tilt of 0.012π for Riamb = 0.225 and less than0.0005π for Riamb = 0.249). The reason that the CL region affects the flow like asingle kernel at its centre is partly due to the strong vorticity amplitude decay awayfrom the CL centre and partly due to the strong vorticity phase tilting which causesthe top and bottom parts of the CL to largely cancel each other’s influence at adistance.

Looking at a schematic illustration of a three-kernel representation of theconfiguration of figure 9, drawn in figure 10, it is clear that if the induced verticalvelocity field from the two kernels is shifted by a quarter wavelength, symmetry yieldsthat the maximum induced displacement at the CL should be one eighth wavelength inbetween the two IWs. Then the vorticity anomaly at the CL, generated in quadraturein response to the induced displacement, induces a vertical velocity anomaly, whichtends to oppose the existing displacement anomaly at all levels. Since this CL-inducedvertical velocity is not exactly in anti-phase with the IW displacements, it also tends


Length

Hei

ght

−0.3

−0.5

−0.7

−0.5

−0.3

−0.1

0

−0.6

−0.4

−0.2

0

0.7

0.5

0.3

0.1

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8

–0.001

0

0.001

−0.01

4

−0.0

1

−0.0

18

−0.0

06

−0.0

02

−0.014−0.01

−0.006

−0.002

0 0

0.0180.018

0.014

0.01

0.008

–1.5

–1.0

–0.5

−0.014

−0.01

−0.0

06

−0.0

02

−0.018

0 0

0.018

0.014

0.01

0.00

6

0.00

2

0.5

1.0

1.5

(0.0015)

(−0.0015)

Criticallayer

Figure 9. The x–z contour plots of the vorticity and the displacement for k = 1.07 andRiamb = 0.249. The colour contours show q with the positive values in red and yellow and thenegative values in blue and green. The black contours show ζ , where the lighter shades arethe negative values and the darker shades are the positive values. The normalized q contourinterval is 0.2, and the normalized ζ values are marked on the plot. The CL is enlarged andoverlain on the centre so that the height axis is discontinuous at z = ± 0.5. The length is givenin units of wavelength.

to shift the IW in the direction of the shear (a hindering effect). Moreover, theCL-induced displacement anomaly also acts to decrease its own amplitude growth ina standing-wave-like fashion.

As Riamb approaches a quarter, the vorticity at the CL approaches infinity, and itsinduced destructive velocity completely masks the constructive interference betweenthe two IW vorticity delta functions. This argument is somewhat circular for normalmodes, since as the growth rate goes to zero, (2.2) indicates that the CL vorticity goesto infinity; therefore it is difficult to differ between cause and effect. For this reasonwe simulate the evolution towards the normal modes from various initial conditions,including beginning from the phase-locked IWs with zero ambient vorticity. Thesimulations (not shown here) indicate indeed that the CL is the region which ispersistently amplified by the IWs, and as the CL vorticity grows it acts to dampen theIWs. The unstable mode which last survives is that for which the IWs are in the optimalconstructive interference phase of π/2 (which is of wavenumber k = 1.07, shown infigures 8–10). As Riamb crosses a quarter all modes become neutral. This schematicpicture is slightly altered by the integrated effect of the non-zero vorticity kernelsoutside the CL and the interfaces. These kernels are in phase with their adjacentinterfaces (figure 8) and therefore act to help them to counter-propagate against theshear. This effect overwhelms the opposed hindering by the CL, and consequentlythe unstable regime is shifted towards the helping regime in the dispersion relationas Riamb increases (figure 7). The other difference is that when Riamb < 1/4 the CL is


q ~ –i • ς

q ~ – ς

q ~ ς

Z = 1

Z = –1

Z = 0

–

– + –

+ – + –

+

+–

+

U–

U–

Figure 10. The three-wave mechanism for the most unstable wavenumber resulting in aπ/2 phase difference between the interfacial kernels’ displacement. The wavy lines illustratethe displacement, and the circles show the vorticity with the anomaly sign marked and thenegative circles dashed. The large arrows represent the self-vertical velocity generated by thelocal wave, while the small dashed or dotted arrows are the induced vertical velocity generatedby the two other levels. The IWs (z = ±1) are pure ζ − and ζ+ kernels respectively as in thetwo-density-jump model. The CL wave (z = 0) consists of q and ζ anomalies which are π/2out of phase and a displacement phase difference of π/4 from the IWs so that it is situated inequal spacing from both.

actually a layer with a finite width (figure 9). Nonetheless its overall effect remainsdestructive.

4.3. Energy and momentum flux consideration

The destructive role of the CL seems surprising in light of the fact that this is aplace where energy is extracted from the mean flow to the perturbation (Lindzen &Tung 1978). For the barotropic/baroclinic case Harnik & Heifetz (2007) showed thatthe CRW kernels, adjacent to the CL, are governed by the action-at-a-distance fromremote dominating kernels. Next we show that this is also the case for the stablystratified shear set-up. Lindzen & Tung (1978) and Lindzen & Barker (1985) lookedat the energy flux carried by propagating waves (across the shear) at the vicinity ofthe CL and found that energy flux diverges at the CL when over-reflection occurs.To inspect the energy flux they applied the first theorem of Eliassen & Palm (1960)(strictly valid for steady flow, but has been shown to be a good approximation forsmall growth rate instability as well):

pw = −ρ0(U − c)uw (4.3)

(where the bar denotes zonal averaging). For positive shear, energy flux divergence(pw > 0 above the CL and pw < 0 below it) is obtained when the momentum fluxuw < 0 at both its sides (and thus the energy growth −uwUz is positive there). Thisview of the CL and energy extraction from the mean flow arises naturally in the over-reflection view of things. Here we view this energy flux divergence at the CL throughour kernel action-at-a-distance approach to gain further insight into the relative


– + –

– + –

– + –

– + –

(a) (b)z

U–

Figure 11. The momentum flux (uw) induced by two vorticity anomalies: (a) tilted againstthe shear and (b) tilted with the shear with a phase difference of π/2. The thick arrows are thelocal velocities (vertical and horizontal), while the dotted and dashed arrows are the inducedvelocities by the upper and lower (respectively) perturbations. In between the two anomaliesuw is negative for (a) and positive for (b). Notice that for both (a) and (b), above the topanomaly and below the bottom one the two contributions to uw are of equal magnitude andopposite signs, resulting in a zero total momentum flux.

contribution of the CL and IWs. Towards this end we first develop an understandingof the kernel action-at-a-distance contribution to the momentum flux uw.

By definition, the zonal and vertical velocities induced by a single kernel are inquadrature and therefore do not contribute to the momentum flux uw. Therefore,from the KGW perspective, the momentum flux is inherently non-local, as it resultsfrom the interaction between different kernels. If we denote the contribution to themomentum flux at some level z, from two kernels located at z′ and z′′, as [uw(z′, z′′)]z,then if z is sandwiched between the kernels, i.e. z′′ > z > z′, figure 11 indicates that themomentum flux is negative (positive) when the kernels’ vorticity is tilted westward(eastward) with height. Furthermore, the kernels’ contributions are even in the sensethat [uw(z′, z′′)]z = [u(z′)w(z′′)]z + [u(z′′)w(z′)]z = 2[u(z′)w(z′′)]z. Moreover [uw(z′, z′′)]is independent of height everywhere between the two kernels (if z, for instance, is closerto the upper kernel, from below the weaker contribution from the lower kernel iscompensated by the stronger contribution from the upper one). In contrast, outside thekernels (z > z′′ or z < z′) the contribution vanishes since [u(z′)w(z′′)]z = −[u(z′′)w(z′)]z.

Applying the cases of figure 11 to our problem, as presented for example infigures 8–10, it is clear that the interaction between the two IWs (tilted westward,against the shear) yields a negative contribution to uw in between them and hence apositive contribution to the energy growth. Thus, the mutual amplification betweenthe two kernels can also be expressed by a positive contribution to the energy fluxdivergence at the CL. The kernel at the CL is tilted with the shear relative to bothIWs, and therefore the interaction between the IWs and the CL contributes towardspositive uw and hence towards energy flux convergence at the CL. Furthermore,since the vorticity at the CL (having a small finite width) is tilted with the shear itcontributes towards a positive momentum flux.


In Appendix D we derive the overall integrated expression for the momentum flux,evaluated at height z, in terms of kernels’ interaction (at height z′ and z′′) to be

−[uw]z =1

4

∫ zz′=−∞

∫ ∞z′′=z

Q(z′)Q(z′′) sin[α(z′′) − α(z′)] exp[−k(z′′ − z′)] dz′ dz′′. (4.4)

As a quantitative example we compute uw, for the solution k = 1.07 and Riamb =0.24,at z = 0. For this case the CL is concentrated at |z|


approach has been under investigation more intensely and therefore covers manycases such as viscous flow (Lindzen & Rambaldi 1986) and normal-mode as wellas non-modal instabilities (Lindzen & Rosenthal 1981). Non-normal-mode instabilityhas only been fairly recently interpreted by the field perspective (Heifetz & Methven2005).

As opposed to the Eady (1949) model, where the Rossby IWs are generated bypotential vorticity gradients, here the vorticity IWs are gravity waves generated bybuoyancy gradients. In the Eady model each interface supports one CRW, whereashere two gravity waves exist at each density jump, one counter-propagating and onepro-propagating with the mean flow. Although this complicates the dynamics andalthough the wave generation mechanism is different, we have shown in detail thatthe essence of the instability is the same as in the Eady model. The two counter-propagating waves, one from each interface, dominate the dynamics as they phaselock each other in a growing configuration. The role of the pro-propagating waveswas found to be minor (mainly in expanding the spectral range of instability in thelow-wavenumber hindering regime). These results are in line with the view of Baines& Mitsudera (1994).

This simple set-up by itself emphasizes the subtleties associated with the M–Hcriterion. A single density jump in a plane Boussinesq Couette flow satisfies the M–Hnecessary, but insufficient, condition for instability (since the Richardson number is adelta function there and zero elsewhere); nonetheless the flow is neutral. Only whenanother density jump is added so that gravity waves can interact constructively at adistance, the flow becomes unstable. Furthermore, it seems strange that if we add anambient stratification in between the density jumps so that the ambient Richardsonnumber exceeds the small value of a quarter (compared with infinity in the two jumps),the instability is shut down. The explanation of Baines & Mitsudera (1994), that thewaves are then absorbed by the CL and hence are heavily damped, mixes argumentsfrom the two perspectives. Here we used the field theory KGW formulation derivedby Harnik et al. (2008), which can be applied to continuous stratification as well, toshow that the CL is the region which is persistently amplified by the IWs, simplybecause fluid parcels there have no relative zonal motion with respect to the lockedIWs. Hence, although the ambient stratification is small and uniform everywhere, atthe CL the particle displacement anomalies are large so that the resulting buoyancygeneration of vorticity is also large. The resulting continuous dynamical structure canbe then simplified to a three-kernel dynamics – two counter-propagating IWs at theinterfaces and one standing-wave kernel at the CL.

From this perspective, the CL does not ‘absorb’ the two interacting waves; ratherit is a standing-wave-like kernel which decreases the two mutually amplifying kernels.This instability is allowed only when the CL effect is weak enough, that is, when theRichardson number is smaller than a quarter. This destructive role of the CL mayseem surprising, when recalling that in the CL energy flux is diverging; hence energyis being yanked from the mean flow to the perturbation. The kernel formulationprovides a rigorous way to compute the action-at-a-distance contribution of theremote kernels to the energy flux. It then turns out that it is the interaction betweenthe two IWs that contributes towards positive energy flux divergence at the CL, whilethe interaction between the CL kernel and the IWs contributes towards energy fluxconvergence and therefore towards stabilization of the flow (a similar effect has beenobserved by Harnik & Heifetz 2007 in barotropic and baroclinic shear flows).

To conclude, we investigated one of the simplest set-ups for stably stratified shearflow instability, and still the physical mechanism of the instability is far from being


0 0.2 0.4 0.6 0.8 1.0 1.2 1.4−0.10

−0.05

0

0.05

0.10

(a)G

row

th r

ate

(kc i

)Tanh profileTwo-jump profile

−1.5 −1.0 −0.5 0 0.5 1.0 1.5−0.5

0

0.5

1.0

1.5

(b)

Height

Wavenumber (k)

Am

plit

ude

and

phas

e/π

Amplitude for tanh profilePhase for tanh profileRi for tanh profileAmplitude for two-jump profilePhase for two-jump profile

Ri = 0.5Ri = 1

Ri = 2Ri = 1.5

Figure 12. The continuous ‘tanh’ density profile solution in comparison with thetwo-density-jump solution for (a) the growth rate as a function of the wavenumber and

(b) the amplitude and phase for the most unstable wavenumber in the case of R̂i = 1. The tanhsolution converges to the two-density-jump solution when → 0. In (a), = 0.01 was chosen,while for (b), = 0.1, since the former value produces indistinguishable amplitude solutions(the vorticity amplitude for the tanh solution appears to be two delta functions).

simple. A possible reason for this is that the M–H theorem is necessarily spectral(one must assume a priori a modal solution to obtain the condition) and hence is notderivable from Hamiltonian considerations (Shepherd 1990). This stands in contrastto barotropic and baroclinic shear flows whose instability conditions can be derivedfrom globally conserved Hamiltonians (Heifetz, Harnik & Tamarin 2009).

We are grateful to the Israeli Science Foundation (1084/06 and 1370/08).

Appendix A. Two density jumps as a limit case of a continuous density profileThe continuous density profile (figure 2b)

ρ = ρ0 −|�ρ|

2

[tanh

z − h

h

+ tanhz + h

h

](A 1)

yields the Richardson number

Ri =R̂i

2

[2 − (tanh2 ((z − 1)/) + tanh2 ((z + 1)/))], (A 2)

where > 0 is a tunable parameter measuring the relative width of the layer (inwhich the density varies) compared with the distance between the layers. When

→ 0, (A 1) and (A 2) converge respectively to (3.1) and (3.3). It is straightforwardto show that the M–H criterion is satisfied for this continuous profile. In figure 12


we take = 0.01 and solve the eigenvalue problem, both by the kernel approach tothe vorticity–displacement equations (Appendix B) and by a shooting method of theTaylor–Goldstein equation. Figure 12(a) shows the growth rate solution for various

values of R̂i. The agreement with the discontinuous solution (figure 5a) is excellent.

Figure 12(b) shows the profile of the Richardson number for R̂i = 1 and the vorticitystructure of its most unstable mode (k = 0.43). The vorticity is maximized at z = ± 1,and the phase difference between these two points ( ≈ 1.4π) is in perfect agreementwith the discontinuous profile (the plots second from the bottom in figure 5c,d ).

Appendix B. Numerical modal solution of the vorticity–displacement equationsDiscretizing the normalized version of (2.2) and (2.4) yields(

∂

∂t+ ikzj

)qj = −ikRij ζj , (B 1)

(∂

∂t+ ikzj

)ζj =

N∑n=1

qnGj,n. (B 2)

The domain is taken between (−3.5, 3.5) to assure no numerical boundary effect(the density jumps are located at z = ±1). Furthermore, N = (7/�z) + 1 =1751 isthe number of points at which sufficient accuracy is obtained for �z = 4 × 10−3.The Richardson number profile of (4.2) is discretized so that Ri(z = ±1) = R̂i/�zand Ri(z �= ±1) = Riamb elsewhere. The discretized Green function is Gj,n = −�z ·(i/2) exp(−k|zj − zn|). Redefining G̃= (i/k)G, (B 1) and (B 2) can be written in thematrix form:

∂

∂t

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

q1

··

qN

ζ1

··

ζN

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠2N×1

= −ik ·

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

z1 0 · 0 Ri1 0 · 00 · · 0 · ·· · 0 · · 00 · 0 zN 0 · 0 RiN

G̃1,1 · · G̃1,N z1 0 · 0· · · 0 · ·· · · · · 0

G̃N,1 · · G̃N,N 0 · 0 zN

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠︸︷︷︸

A2N×2N

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

q1

··

qN

ζ1

··

ζN

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠2N×1

, (B 3)

where the normal modes are the eigensolution of (B 3).

Appendix C. Analytical solution of Taylor–Goldstein equationApplying the linearized material derivative D/Dt to the normalized version of

(2.2), expressing the vorticity and the vertical velocity in terms of the streamfunctionand looking for the normal-mode solution yield the Taylor–Goldstein equation (e.g.Baines 1998) with zero shear curvature:

ψzz +

[Ri

(z − c)2 − k2

]ψ = 0. (C 1)


For the transformation of variables y = (z − c)k and ψ̃ = ψ/√y, the Taylor–Goldsteinequation is transformed into the modified Bessel equation (e.g. Sneddon 1956)

y2ψ̃yy + yψ̃y − (y2 + ν2)ψ̃ = 0, (C 2)

with ν =√

1/4 − Ri, whose general solution is

ψ =√

y[C1Iν(y) + C2I−ν(y)], (C 3)

where Iν and I−ν are the modified Bessel functions and C1 and C2 are two constants.Equivalently we can obtain the Bessel function from the Taylor–Goldstein equationby writing ỹ = iy,

ỹ2ψ̃ỹỹ + ỹψ̃ỹ + (ỹ2 − ν2)ψ̃ = 0, (C 4)

whose solution is

ψ =√

y[AH1(iy) + BH2(iy)], (C 5)

where H1 and H2 are the Hankel functions and A and B are two constants. Since thetwo formulations are equivalent we arbitrarily choose to use the Hankel functions forthe matching solution.

We hence decompose the set-up of figure 2(c) into the three domains, namely (i)z > 1, (ii) −1


Hence, in order to obtain a non-trivial solution, Det(M) = 0, where

M =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

H1(iy1) −H1(iy1)

∂

∂z[√

yH1(iy)]y1 +R̂i

(1 − c)2√

y1H1(iy1) −∂

∂z[√

yH1(iy)]y1

0 H1(iy2)

0∂

∂z[√

yH1(iy)]y2

−H2(iy1) 0

− ∂∂z

[√

yH2(iy)]y1 0

H2(iy2) −H2(iy2)∂

∂z[√

yH2(iy)]y2 −∂

∂z[√

yH2(iy)]y2 +R̂i

(1 + c)2√

y2H2(iy2)

⎞⎟⎟⎟⎟⎟⎟⎟⎠. (C 13)

This yields the phase speed c. The coefficients A1, A2, B2 and B3 are then obtainedwhen solving (C 10) and (C 11). After deriving the solution for ψ we easily obtain thevertical displacement ζ = ψ/(z − c) and the vorticity q = −[Ri/(z − c)2]ψ .

Appendix D. Integrated kernel contribution to the momentum fluxWe wish to express

uw =1

2π

∫ 2π0

Re(u)Re(w)d(kx) (D 1)

in terms of the inversion of the vorticity profile q = Q(z) exp{i[kx + α(z)]}. For clarity,we redefine the Green function to be the real, positive-definite and non-dimensionalevanescent function, G̃(z, z′) = iG(z, z′) = (1/2) exp(−k|z − z′|), and use (2.3) to write

Re(w) =

∫z′

Q(z′) sin[kx + α(z′)]G̃(z, z′) dz′, (D 2)

where the sine in the integrand simply indicates that the induced vertical velocityis located a quarter of wavelength to the east of the inducing vorticity anomaly(figure 11). Similarly we can write that

Re(u) = Re

(i

k

∂w

∂z

)=

1

k

∫z′′

Q(z′′) cos[kx + α(z′′)]∂

∂zG̃(z, z′′) dz′′. (D 3)

Since (∂/∂z)G̃(z, z′′) > 0 for z < z′′ and (∂/∂z)G̃(z, z′′) < 0 for z > z′′, the integrandindicates that the induced zonal velocity anomaly below (above) the inducing vorticityanomaly is in phase (anti-phase). Substituting (D 2) and (D3) into (D 1) we obtainafter zonal integration

uw =

∫z′

∫z′′

1

2kQ(z′)Q(z′′) sin[α(z′) − α(z′′)]G̃(z, z′) ∂

∂zG̃(z, z′′) dz′dz′′. (D 4)

It is clear from the integrand that a single kernel (where z′ = z′′) does notcontribute to the momentum flux. Writing explicitly the Green function and its


derivative,

G̃(z, z′) =1

2

{exp[−k(z − z′)] for z > z′,exp[k(z − z′)] for z < z′ ,

∂

∂zG̃(z, z′′) =

k

2

{− exp[−k(z − z′′)] for z > z′′,exp[k(z − z′′)] for z < z′′ .

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭(D 5)

Then if z′′ > z > z′, the integrand yields −(1/8)Q(z′)Q(z′′) sin[α(z′′) −α(z′)] exp[−k(z′′ − z′)], which is independent of z and positive if the vorticityis tilted westward with height. Flipping between the dummy variables z′ and z′′

yields the same contribution, indicating that indeed [u(z′)w(z′′)]z = [u(z′′)w(z′)]z. If,however, z > z′′ >z′ or z < z′


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Journal of Fluid Mechanics Vorticity inversion and …harnik/publications/Rabinovich_etal_11... · 2014. 4. 22. · 304 A. Rabinovich, O. M. Umurhan, N. Harnik, F. Lott and E. Heifetz